Type: Seminar

Distribution: World

Expiry: 10 Oct 2008

CalTitle1: Algebra Seminar: Messing -- p-adic Hodge theory and p-adic periods

Auth: anthonyh@bari.maths.usyd.edu.au

Classical Hodge theory deals with projective and smooth algebraic varieties defined
over C. If X is such a variety defined over the subfield Q, then associated
to X are two cohomology theories:
- The Betti cohomology of X, denoted H*
_{B}(X). It is defined as the singular cohomology of the underlying space X(C) of the complex points of X with coefficients in Q. - The algebraic de Rham cohomology of X, denoted H*
_{dR}(X). This is defined as the hypercohomology of the de Rham complex Omega*_{X/Q}.
_{dR}(X) and
H*_{B}(X) after complexification. If we choose bases for these two
Q-vector spaces, this isomorphism is given by an element in GL_{N}(C),
called the period matrix. In general little is known about the entries in this matrix,
although the degree of transcendence of the field generated over Q by its entries was
conjectured by Grothendieck to have dimension equal to that of a reductive algebraic
group defined over Q, the Mumford-Tate group associated to X.
We will discuss the p-adic analogues of this classical situation.
Beginning with the work of Barsotti, Tate, Grothendieck during the 1950's and 1960's,
the theory of p-divisible groups developed. In his Nice ICM talk, Grothendieck raised
the problem of the "mysterious functor". In 1978 Fontaine introduced a ring
B After the seminar we will take the speaker to lunch. See the Algebra Seminar web page for information about other seminars in the series. Anthony Henderson anthonyh@maths.usyd.edu.au. |